"How
I need

a
drink,

alcoholic
of

course,
after

the
heavy

lectures

involving

quantum

mechanics..."

Underground
Pi

Digging
Out from Under Pseudoscience

By
Mark Cowan

In
the Washington Park Station of the Westside Light Rail Tunnel, in the 16-million
year recitation of history that runs alongside a 260-foot core sample of
Portland’s West Hills, you will find, etched in granite, the first 107
digits of the transcendental constant pi (figure 1).

Figure
1. Pi as it appears in the Washington Park Station of the Westside MAX
Light Rail.

Pi, as every schoolchild knows, is the ratio of the circumference of a
circle to its diameter - one of the fundamental ratios of the space that
fills our universe. Its digits never repeat, and they follow no pattern.
Other than its role in geometry, and the fact that it pops up in many diverse
branches of mathematics, pi has no special significance. It does not mean
anything...so far as we know.

Yet pi has exerted a steady pull on the human imagination. The Babylonians
and Egyptians knew its value to within a half a percent some 4000 years
ago. By the 3rd century BC, Archimedes had rectified the circle, nearly
invented the calculus, and established pi’s value to about one part in
100,000 by the use of regular polygons. And by the 5th century AD a Chinese
father and son, using a variation of this method, pinned down eight digits-a
precision unequaled in Europe until the 16th century. Their laborious extraction
of square roots was aided by the early Chinese introduction of a blank
for zero.

By diligent use of Archimedes’ method, in a 1596 paper entitled “On the
Circle,” Dutch mathematician Ludolph van Ceulen single-handedly delivered
the first 20 digits - then challenged anybody to top it. None did, but
he soon extended his claim on history first to 32, and then to 35 digits
(figure 2), of which the last three were engraved on his tombstone. To
this day pi remains “the Ludolphine number” in Germany.

3.1415926535897932384626433832795028

Figure
2. Pi as per van Ceulen (1596).

But the big guns were ready to fire. The methods known since Archimedes’
time could, theoretically at least, calculate pi to any desired degree
of accuracy, the only limits being the calculator’s fortitude. When European
mathematics began to flourish, the methods themselves were improved.

In 1665 and 1666, during the Plague, Issac Newton developed the calculus-and
offhandedly produced an efficient infinite series for calculating pi .
Evaluating only 22 terms of the series yielded 16 digits. He saw no practical
value to this effort, however, and later apologized for how far he had
carried his computation “Having,” as he wrote, “no other business at the
time.”

But if a giant like Newton could fall under pi’s transcendental spell just
for lack of anything better to do, was anyone truly immune?

Heedless of such reservations, the hunt continued. With various modifications
to improve efficiency, by 1719 the French mathematician de Lagny had sweated
his way through 127 decimal digits (figure 3), a record that would stand
for 75 years.

Further progress required more efficient tools. Around 1755 Leonhard Euler,
perhaps the greatest mathematician of all time, discovered the fastest
converging series yet known. Using it, he worked out the first 20 decimal
places of pi in a single hour. But, doubtless mindful of the limited value
of this pursuit, like Newton he went no further. Others, of course, were
more than willing to extend the tally-and naturally they used his methods.

But one wonders: if any of these early pi hunters were somehow to wander
down the Washington Park MAX Station today, what might they think of those
107 digits etched in cold granite? Would Newton sneer? Would Euler wince?

For the physical accuracy implied by 106 decimal places of pi has no counterpart
in reality. If you inscribed a circle the diameter of the known universe
(which has varied recently, but we’ll use 24-billion light-years), and
then calculated its circumference by use of those 106 digits, the error
due to truncation would be 1/1061 of the width of an atomic nucleus!

Still the hunt went on. Calculating prodigy Johann Dase produced 200 decimal
digits of pi in just two months in 1844, with others ringing up slightly
larger tallies - until finally William Shanks published 707 digits of pi
in 1873-74. This record stood until 1945-when he was shown to have gotten
the last 180 digits wrong.

But now the electronic computer was on the horizon, and by 1949 ENIAC had
churned out 2037 places in 70 hours. The digital floodgates opened. Pi’s
current world record now stands at 51,539,600,000 decimal digits set in
June of 1997 by Kanada and Takahashi (1) at the Tokyo
Computer Centre after 29 hours on a machine with 1024 processors. That’s
61 million times faster than ENIAC per digit. Interestingly enough, the
two digits beginning at position 49,999,999,999 in both pi and 1/pi are
42.

Ivars Peterson’s online Mathland delivers more information on pi
mania - which, of course, continues unabated - with Internet links to get
you started (2). There you can marvel at people who
memorize great hunks of pi . You can also learn of a new formula that delivers
specific hexadecimal digits of pi -without knowing any of the preceding
ones! This completely unanticipated result is being put to use to calculate,
via an Internet network (3), both the 5 and 40 trillionth
binary digit of pi . No equivalent formula yet exists for decimal digits.
Seems like there’s an argument against creation in there somewhere...

If, by now, you just can’t live without your own big piece of pi, running
Piw131 (4) overnight on a decent PC with 32 megabytes
or more of memory will get you a cool million digits by morning. If that’s
not enough, you can search (5) the first 50 million
digits for any string of numbers up to 127 digits. But consider this: in
about an hour I wrote, from scratch, a simple program (6)
that computes the circumference of a unit circle using nothing more advanced
than square roots. Run under QBASIC it delivers Newton’s 16 digits after
only 26 iterations.

Sure, it was fun to do - but is there any real point to any of this, after
all?

Well, that’s where it gets interesting. The distribution of digits in the
first 50 billion digits of pi is statistically normal (6).
But a recent study (7) has found that the distribution
of repeating strings of digits is not. And nobody knows why that should
be so! So pi , it would seem, still contains some curious implications
for number theory. And the digital expression of it is the source of a
new kind of mathematical analysis.

Which brings us, uhm, full circle-and back to “Pi Underground.”

According to Rebecca Banyas of Tri-Met’s Westside Light Rail, the artist,
Bill Will, “got his information on pi from a reference book called The
History of Pi(8). The numbers that appear on
the wall are the same as those in the book.”

Well, sort of. You may have already noticed, however, a slight discrepancy
between the values carved into the tunnel and those worked out nearly 300
years earlier by de Lagny. This discrepancy was first spotted by a MAX
engineer who had memorized pi to 12 places as a child. But the reason for
the error remained obscure. Was it Art? A bad job of typesetting? Deliberate?
Just to see if anybody was paying attention?

After I searched strings of the Washington Park Station digits against
the half-million pi digits on my computer, the source of the error became
clearer. And checking out a copy of A History of Pi made it obvious.
Artist Bill Will wasn’t taking liberties with a constant of the universe.
He was just unfamiliar with the format of mathematical tables (figure 4).

Figure
4. The first 2000 of the 10,000 decimal places of pi as printed in A
History of Pi. Note that the artist used the digits going down the columns
rather than across the rows.

Of course, most people aren’t going to notice - or probably much care.
But if you spot somebody standing around in the tunnel, reciting something
that begins “How I need a drink...,” at least you’ll be able to chime in,
“alcoholic of course...”(9).

And they’ll think you’re both loonies, and won’t let either of you on the
train.

REFERENCES
AND NOTES

1. Details
(not
the result, though!). Some statistical analysis of the first 50 billion
digits is provided.

7. Unfortunately
I’ve been unable to find where I saw this; you’ll have to trust me. :)

8. Beckmann, Petr.
1971. A History of Pi. The Golem Press. I owe a great debt to this
excellent book and have drawn much from it for this article.

9. “How I need a
drink, alcoholic of course, after the heavy lectures involving quantum
mechanics” gives you 15 digits. Other mnemonics exist, including a poem
somewhat reminiscent of Poe’s The Raven that delivers 740 digits.
See (2).